DL0001

matrix_recursive_node_code_exists

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Six explicit doubled-Cantor constructors code one exact matrix evaluation record with conservatively shared intermediate values.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Exact expanded first-order arithmetic statement

forall d pb pc nb nc p n. exists z. (exists mdr_a_code_exists mdr_b_code_exists mdr_c_code_exists mdr_e_code_exists mdr_f_code_exists. ((mdr_a_code_exists = ((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) /\ ((mdr_b_code_exists = ((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))) /\ ((mdr_c_code_exists = ((mdr_a_code_exists) + (mdr_b_code_exists)) * S ((mdr_a_code_exists) + (mdr_b_code_exists)) + ((mdr_b_code_exists) + (mdr_b_code_exists))) /\ ((mdr_e_code_exists = ((p) + (n)) * S ((p) + (n)) + ((n) + (n))) /\ ((mdr_f_code_exists = ((nc) + (mdr_e_code_exists)) * S ((nc) + (mdr_e_code_exists)) + ((mdr_e_code_exists) + (mdr_e_code_exists))) /\ ((z) = ((mdr_c_code_exists) + (mdr_f_code_exists)) * S ((mdr_c_code_exists) + (mdr_f_code_exists)) + ((mdr_f_code_exists) + (mdr_f_code_exists)))))))))

Constructive proof overview

Generated structural guide

Six explicit doubled-Cantor constructors code one exact matrix evaluation record with conservatively shared intermediate values.

The unchanged tactic script uses 0 declared prerequisites and contains 24 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

none

Direct dependents

DL0006 matrix_recursive_record_append

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

24 script commands · 12 reading checkpoints · 0 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

01Fix variables and assumptionsL1–7

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro d
  2. L2
    intro pb
  3. L3
    intro pc
  4. L4
    intro nb
  5. L5
    intro nc
  6. L6
    intro p
  7. L7
    intro n
02Construct an explicit witnessL8–13

Supply the displayed value, then prove that it has the required property.

  1. L8
    exists ((((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb) · expand full local formula (1,480 characters)exists ((((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) * S ((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) + ((((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))))) + (((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) * S ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) + ((((p) + (n)) * S ((p) + (n)) + ((n) + (n))) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))))) * S ((((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) * S ((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) + ((((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))))) + (((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) * S ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) + ((((p) + (n)) * S ((p) + (n)) + ((n) + (n))) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))))) + ((((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) * S ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) + ((((p) + (n)) * S ((p) + (n)) + ((n) + (n))) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n))))) + (((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) * S ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) + ((((p) + (n)) * S ((p) + (n)) + ((n) + (n))) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n))))))
  2. L9
    exists ((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))
  3. L10
    exists ((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))
  4. L11
    exists ((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) * S ((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) + ((((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))))
  5. L12
    exists ((p) + (n)) * S ((p) + (n)) + ((n) + (n))
  6. L13
    exists ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) * S ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) + ((((p) + (n)) * S ((p) + (n)) + ((n) + (n))) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n))))
03Separate the logical casesL14–14

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L14
    split
04Calculate and transport equalitiesL15–15

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L15
    refl
05Separate the logical casesL16–16

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L16
    split
06Calculate and transport equalitiesL17–17

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L17
    refl
07Separate the logical casesL18–18

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L18
    split
08Calculate and transport equalitiesL19–19

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L19
    refl
09Separate the logical casesL20–20

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L20
    split
10Calculate and transport equalitiesL21–21

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L21
    refl
11Separate the logical casesL22–22

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L22
    split
12Calculate and transport equalitiesL23–24

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L23
    refl
  2. L24
    refl

Library-wide reading audit

Original exact command ledger · 24 lines
  1. 0001intro d
  2. 0002intro pb
  3. 0003intro pc
  4. 0004intro nb
  5. 0005intro nc
  6. 0006intro p
  7. 0007intro n
  8. 0008exists ((((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) * S ((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) + ((((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))))) + (((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) * S ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) + ((((p) + (n)) * S ((p) + (n)) + ((n) + (n))) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))))) * S ((((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) * S ((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) + ((((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))))) + (((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) * S ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) + ((((p) + (n)) * S ((p) + (n)) + ((n) + (n))) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))))) + ((((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) * S ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) + ((((p) + (n)) * S ((p) + (n)) + ((n) + (n))) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n))))) + (((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) * S ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) + ((((p) + (n)) * S ((p) + (n)) + ((n) + (n))) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n))))))
  9. 0009exists ((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))
  10. 0010exists ((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))
  11. 0011exists ((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) * S ((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) + ((((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))))
  12. 0012exists ((p) + (n)) * S ((p) + (n)) + ((n) + (n))
  13. 0013exists ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) * S ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) + ((((p) + (n)) * S ((p) + (n)) + ((n) + (n))) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n))))
  14. 0014split
  15. 0015refl
  16. 0016split
  17. 0017refl
  18. 0018split
  19. 0019refl
  20. 0020split
  21. 0021refl
  22. 0022split
  23. 0023refl
  24. 0024refl